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"Bonds will form only if the potential energy will be lower when the atoms are bonded than when they are separate."

Is this statement true? If so, how do endothermic reactions exist? Can't you add energy to a system to make the bonds form albeit unfavorably?

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Two different factors play a role when considering a reaction. The energy of the bonds etc. (enthalpy) and the entropy. Together, these two components make up what is called the 'Gibbs Free Energy'. (Wikipedia)

The change in Gibbs free energy ($\Delta{}G$) of a system must be $< 0$ for a reaction to be spontaneous. From this it is possible to see that a reaction where the entropy increases drastically, but the enthalpy increases. In this case a reaction can be endothermic.

A case of terminology: The 'more correct' term (my teacher always said) is exergonic or endergonic. What this means is that the reaction has a $\Delta{}G < 0$ or $\Delta{}G > 0$ respectively.

As @Cohorent mentioned before, it is also possible to have a system that is larger than just the chemical reaction (adding a battery for example). Meaning that the reaction is maybe endergonic, but a different energy source is used to drive the reaction. Note then, that the $\Delta{}G$ for the entire system is still below zero.

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They only form of the overall energy of the process is lower. You can form a less stable bond if you put in energy either through the formation of another bond which is more stable or through energy in form of light or electricity.

Example: you can electrochemical my form aluminum from aluminum oxide.

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